# Properties

 Label 475.2.a.e Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.79289409601$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 2) q^{8} + \beta_1 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (-b2 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 - 2*b1) * q^6 + (b2 - 1) * q^7 + (-2*b2 + b1 - 2) * q^8 + b1 * q^9 $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 2) q^{8} + \beta_1 q^{9} - \beta_{2} q^{11} + ( - \beta_{2} + \beta_1 - 3) q^{12} + (3 \beta_{2} - 3 \beta_1 + 1) q^{13} + ( - \beta_{2} + 2) q^{14} + (\beta_{2} - 2 \beta_1 - 1) q^{16} + ( - \beta_{2} - 5) q^{17} + (\beta_{2} + 3) q^{18} + q^{19} + (2 \beta_{2} - \beta_1 - 1) q^{21} + (\beta_{2} - \beta_1 - 1) q^{22} + (2 \beta_{2} - 3 \beta_1 - 1) q^{23} + 5 q^{24} + ( - 6 \beta_{2} + 4 \beta_1 - 7) q^{26} + (3 \beta_{2} - 2 \beta_1 + 2) q^{27} + ( - \beta_{2} + \beta_1 - 1) q^{28} + ( - \beta_{2} - 2) q^{29} + (3 \beta_{2} - \beta_1 + 1) q^{31} + (\beta_{2} - 2 \beta_1) q^{32} + ( - \beta_{2} + \beta_1 + 2) q^{33} + (\beta_{2} - 6 \beta_1 + 4) q^{34} + ( - \beta_{2} + 2 \beta_1 - 2) q^{36} + ( - 5 \beta_{2} + 2 \beta_1 - 4) q^{37} + (\beta_1 - 1) q^{38} + (2 \beta_{2} + 3 \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{41} + ( - 3 \beta_{2} + \beta_1) q^{42} + ( - 2 \beta_{2} + 6 \beta_1 - 1) q^{43} - q^{44} + ( - 5 \beta_{2} + \beta_1 - 6) q^{46} + ( - \beta_{2} + 5 \beta_1 - 5) q^{47} + (2 \beta_{2} + 3 \beta_1 + 1) q^{48} + ( - 4 \beta_{2} + \beta_1 - 4) q^{49} + (4 \beta_{2} + \beta_1 + 7) q^{51} + (4 \beta_{2} - 7 \beta_1 + 11) q^{52} + (2 \beta_{2} - 2 \beta_1 - 9) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 - 5) q^{54} + (4 \beta_{2} - 2 \beta_1 - 1) q^{56} + ( - \beta_{2} - 1) q^{57} + (\beta_{2} - 3 \beta_1 + 1) q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + ( - 4 \beta_{2} + 4 \beta_1 - 1) q^{62} + q^{63} + ( - 5 \beta_{2} + 5 \beta_1 - 3) q^{64} + (2 \beta_{2} + \beta_1) q^{66} + (\beta_{2} - 4 \beta_1 + 6) q^{67} + ( - 5 \beta_{2} + 5 \beta_1 - 11) q^{68} + (3 \beta_{2} + 4 \beta_1) q^{69} + ( - 2 \beta_1 + 3) q^{71} + (\beta_{2} - 3 \beta_1 + 1) q^{72} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{73} + (7 \beta_{2} - 9 \beta_1 + 5) q^{74} + (\beta_{2} - \beta_1 + 2) q^{76} + (3 \beta_{2} - \beta_1 - 2) q^{77} + (\beta_{2} - 2 \beta_1 + 15) q^{78} + (5 \beta_{2} - 4 \beta_1 - 3) q^{79} + (\beta_{2} - 2 \beta_1 - 6) q^{81} + (7 \beta_{2} - 4 \beta_1 + 15) q^{82} + ( - 6 \beta_{2} - 3 \beta_1 - 2) q^{83} + ( - \beta_1 + 2) q^{84} + (8 \beta_{2} - 3 \beta_1 + 17) q^{86} + (\beta_{2} + \beta_1 + 4) q^{87} + ( - 2 \beta_{2} + \beta_1 + 3) q^{88} + ( - 2 \beta_{2} + 5 \beta_1 - 9) q^{89} + ( - 8 \beta_{2} + 3 \beta_1 + 2) q^{91} + (2 \beta_{2} - 5 \beta_1 + 6) q^{92} + (2 \beta_{2} - \beta_1 - 6) q^{93} + (6 \beta_{2} - 6 \beta_1 + 19) q^{94} + (\beta_{2} + 3 \beta_1) q^{96} + (3 \beta_{2} + \beta_1 + 5) q^{97} + (5 \beta_{2} - 8 \beta_1 + 3) q^{98} + ( - \beta_1 - 1) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (-b2 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 - 2*b1) * q^6 + (b2 - 1) * q^7 + (-2*b2 + b1 - 2) * q^8 + b1 * q^9 - b2 * q^11 + (-b2 + b1 - 3) * q^12 + (3*b2 - 3*b1 + 1) * q^13 + (-b2 + 2) * q^14 + (b2 - 2*b1 - 1) * q^16 + (-b2 - 5) * q^17 + (b2 + 3) * q^18 + q^19 + (2*b2 - b1 - 1) * q^21 + (b2 - b1 - 1) * q^22 + (2*b2 - 3*b1 - 1) * q^23 + 5 * q^24 + (-6*b2 + 4*b1 - 7) * q^26 + (3*b2 - 2*b1 + 2) * q^27 + (-b2 + b1 - 1) * q^28 + (-b2 - 2) * q^29 + (3*b2 - b1 + 1) * q^31 + (b2 - 2*b1) * q^32 + (-b2 + b1 + 2) * q^33 + (b2 - 6*b1 + 4) * q^34 + (-b2 + 2*b1 - 2) * q^36 + (-5*b2 + 2*b1 - 4) * q^37 + (b1 - 1) * q^38 + (2*b2 + 3*b1 - 4) * q^39 + (-2*b2 + 5*b1 - 2) * q^41 + (-3*b2 + b1) * q^42 + (-2*b2 + 6*b1 - 1) * q^43 - q^44 + (-5*b2 + b1 - 6) * q^46 + (-b2 + 5*b1 - 5) * q^47 + (2*b2 + 3*b1 + 1) * q^48 + (-4*b2 + b1 - 4) * q^49 + (4*b2 + b1 + 7) * q^51 + (4*b2 - 7*b1 + 11) * q^52 + (2*b2 - 2*b1 - 9) * q^53 + (-5*b2 + 5*b1 - 5) * q^54 + (4*b2 - 2*b1 - 1) * q^56 + (-b2 - 1) * q^57 + (b2 - 3*b1 + 1) * q^58 + (2*b2 + 2*b1 - 2) * q^59 + (b2 + b1 + 1) * q^61 + (-4*b2 + 4*b1 - 1) * q^62 + q^63 + (-5*b2 + 5*b1 - 3) * q^64 + (2*b2 + b1) * q^66 + (b2 - 4*b1 + 6) * q^67 + (-5*b2 + 5*b1 - 11) * q^68 + (3*b2 + 4*b1) * q^69 + (-2*b1 + 3) * q^71 + (b2 - 3*b1 + 1) * q^72 + (-2*b2 + 3*b1 - 2) * q^73 + (7*b2 - 9*b1 + 5) * q^74 + (b2 - b1 + 2) * q^76 + (3*b2 - b1 - 2) * q^77 + (b2 - 2*b1 + 15) * q^78 + (5*b2 - 4*b1 - 3) * q^79 + (b2 - 2*b1 - 6) * q^81 + (7*b2 - 4*b1 + 15) * q^82 + (-6*b2 - 3*b1 - 2) * q^83 + (-b1 + 2) * q^84 + (8*b2 - 3*b1 + 17) * q^86 + (b2 + b1 + 4) * q^87 + (-2*b2 + b1 + 3) * q^88 + (-2*b2 + 5*b1 - 9) * q^89 + (-8*b2 + 3*b1 + 2) * q^91 + (2*b2 - 5*b1 + 6) * q^92 + (2*b2 - b1 - 6) * q^93 + (6*b2 - 6*b1 + 19) * q^94 + (b2 + 3*b1) * q^96 + (3*b2 + b1 + 5) * q^97 + (5*b2 - 8*b1 + 3) * q^98 + (-b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 $$3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24} - 11 q^{26} + q^{27} - q^{28} - 5 q^{29} - q^{31} - 3 q^{32} + 8 q^{33} + 5 q^{34} - 3 q^{36} - 5 q^{37} - 2 q^{38} - 11 q^{39} + q^{41} + 4 q^{42} + 5 q^{43} - 3 q^{44} - 12 q^{46} - 9 q^{47} + 4 q^{48} - 7 q^{49} + 18 q^{51} + 22 q^{52} - 31 q^{53} - 5 q^{54} - 9 q^{56} - 2 q^{57} - q^{58} - 6 q^{59} + 3 q^{61} + 5 q^{62} + 3 q^{63} + q^{64} - q^{66} + 13 q^{67} - 23 q^{68} + q^{69} + 7 q^{71} - q^{72} - q^{73} - q^{74} + 4 q^{76} - 10 q^{77} + 42 q^{78} - 18 q^{79} - 21 q^{81} + 34 q^{82} - 3 q^{83} + 5 q^{84} + 40 q^{86} + 12 q^{87} + 12 q^{88} - 20 q^{89} + 17 q^{91} + 11 q^{92} - 21 q^{93} + 45 q^{94} + 2 q^{96} + 13 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 + q^11 - 7 * q^12 - 3 * q^13 + 7 * q^14 - 6 * q^16 - 14 * q^17 + 8 * q^18 + 3 * q^19 - 6 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 + q^27 - q^28 - 5 * q^29 - q^31 - 3 * q^32 + 8 * q^33 + 5 * q^34 - 3 * q^36 - 5 * q^37 - 2 * q^38 - 11 * q^39 + q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 - 12 * q^46 - 9 * q^47 + 4 * q^48 - 7 * q^49 + 18 * q^51 + 22 * q^52 - 31 * q^53 - 5 * q^54 - 9 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 + 13 * q^67 - 23 * q^68 + q^69 + 7 * q^71 - q^72 - q^73 - q^74 + 4 * q^76 - 10 * q^77 + 42 * q^78 - 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 + 5 * q^84 + 40 * q^86 + 12 * q^87 + 12 * q^88 - 20 * q^89 + 17 * q^91 + 11 * q^92 - 21 * q^93 + 45 * q^94 + 2 * q^96 + 13 * q^97 - 4 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.37720 −0.273891 2.65109
−2.37720 −1.27389 3.65109 0 3.02830 −0.726109 −3.92498 −1.37720 0
1.2 −1.27389 1.65109 −0.377203 0 −2.10331 −3.65109 3.02830 −0.273891 0
1.3 1.65109 −2.37720 0.726109 0 −3.92498 0.377203 −2.10331 2.65109 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.a.e 3
3.b odd 2 1 4275.2.a.bm 3
4.b odd 2 1 7600.2.a.cc 3
5.b even 2 1 475.2.a.g yes 3
5.c odd 4 2 475.2.b.b 6
15.d odd 2 1 4275.2.a.ba 3
19.b odd 2 1 9025.2.a.bc 3
20.d odd 2 1 7600.2.a.bh 3
95.d odd 2 1 9025.2.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.e 3 1.a even 1 1 trivial
475.2.a.g yes 3 5.b even 2 1
475.2.b.b 6 5.c odd 4 2
4275.2.a.ba 3 15.d odd 2 1
4275.2.a.bm 3 3.b odd 2 1
7600.2.a.bh 3 20.d odd 2 1
7600.2.a.cc 3 4.b odd 2 1
9025.2.a.y 3 95.d odd 2 1
9025.2.a.bc 3 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - 3T_{2} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 3 T - 5$$
$3$ $$T^{3} + 2 T^{2} - 3 T - 5$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 4T^{2} + T - 1$$
$11$ $$T^{3} - T^{2} - 4T - 1$$
$13$ $$T^{3} + 3 T^{2} - 36 T - 103$$
$17$ $$T^{3} + 14 T^{2} + 61 T + 79$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} + 8 T^{2} - 9 T - 125$$
$29$ $$T^{3} + 5 T^{2} + 4 T - 5$$
$31$ $$T^{3} + T^{2} - 30 T + 53$$
$37$ $$T^{3} + 5 T^{2} - 74 T - 395$$
$41$ $$T^{3} - T^{2} - 82 T + 155$$
$43$ $$T^{3} - 5 T^{2} - 113 T + 317$$
$47$ $$T^{3} + 9 T^{2} - 64 T - 311$$
$53$ $$T^{3} + 31 T^{2} + 303 T + 905$$
$59$ $$T^{3} + 6 T^{2} - 40 T - 200$$
$61$ $$T^{3} - 3 T^{2} - 10 T - 1$$
$67$ $$T^{3} - 13T^{2} + 169$$
$71$ $$T^{3} - 7T^{2} - T + 47$$
$73$ $$T^{3} + T^{2} - 30 T + 53$$
$79$ $$T^{3} + 18 T^{2} + 17 T - 395$$
$83$ $$T^{3} + 3 T^{2} - 270 T + 131$$
$89$ $$T^{3} + 20 T^{2} + 51 T - 125$$
$97$ $$T^{3} - 13T^{2} + 169$$